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Logarithmically completely monotonic functions and Gurland’s ratio for the gamma function. (English) Zbl 1132.33304
Summary: Two logarithmically completely monotonic functions involving gamma function are proved. As an application, bounds for the ratio \(T(x,y)=\Gamma(x)\Gamma(y)/\Gamma^2((x+y)/2)\) is established, and the corresponding result of J. Kečkić and P. Vasić [Publ. Inst. Math., Nouv. Sér. 11(25), 107–114 (1971; Zbl 0222.33001)]

MSC:
33B15 Gamma, beta and polygamma functions
26A48 Monotonic functions, generalizations
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